Plotting the Quadratic Function Using Parameters a, b and c

🏆Practice plotting functions with parameters

Plotting the Quadratic Function

Plotting the graph of the quadratic function and examining the roles of the parameters a,b,ca, b, c in the function of the form y=ax2+bx+cy = ax^2 + bx + c

The quadratic function has three relevant characteristics:

aa – the coefficient of X2X^2.
bb – the coefficient of XX.
cc – the constant term.

Steps to graph a quadratic function –

  1. Let's examine the parameter aa and ask: Is the function upward or downward facing?
  2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
  3. Let's find the points of intersection with the XX axis by substituting (Y=0Y=0).
  4. Let's draw a coordinate system and first mark the vertex of the parabola.
    Then, let's examine if the function is smiling or crying and mark the points of intersection with the XX axis that we found. Draw accordingly.
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Test yourself on plotting functions with parameters!

einstein

\( y=x^2 \)

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Graphing the quadratic function ausing parameters a, b and c

Let's present the quadratic function and understand the meaning of each parameter in it.
y=ax2+bx+cy=ax^2+bx+c

aa – the coefficient of X2X^2
    - -determines if the parabola will be a maximum or minimum parabola (sad or smiling)   - determines the steepness – its width.

aa must be different from 0.
If aa is positive – the parabola is a minimum parabola – smiling
If aa is negative – the parabola is a maximum parabola – sad
The larger aa is – the narrower the function will be and vice versa.

bb - the coefficient of XX
     can be any number
indicates the position of the parabola along with CC.
determines the slope of the function at the intersection point with the YY axis
(not relevant to the material) 

cc – the constant term
    is not dependent on XX
can be any number
indicates the position of the parabola along with XX
determines the y-intercept
responsible for the vertical shift of the function


Wonderful. Now, let's move on to plotting the quadratic function.

Steps to graph a quadratic function –

  1. Let's examine the parameter aa and ask: is the function upward or downward facing?
  2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
  3. Let's find the points of intersection with the XX axis by substituting (Y=0Y=0).
  4. Let's draw a coordinate system and first mark the vertex of the parabola.
    Then, let's examine if the function is smiling or sad and mark the points of intersection with the XX axis that we found. Draw accordingly.

Let's see an example -

In the following function:
Y=4X2+4x+3Y=-4X^2+4x+3

  1. a=4a=-4, negative. Therefore, the function is downward facing.
  2. Let's find the vertex of the parabola:
    X=442=48=12X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}
    y=4122+412+3y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3
    y=4y=4
    Vertex of the parabola (12,4) (\frac{1}{2}, 4)  
     
  3. Let's find the intersection points with the XX axis
    Substitute
    Y=0Y=0
    and we get:
    4X2+4x+3=0-4X^2+4x+3=0
    X=12,112 X=-\frac{1}{2},1\frac{1}{2} 
     
  4. Draw a coordinate system, mark relevant points, and draw logically:
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Examples with solutions for Plotting Functions with Parameters

Exercise #1

y=x2+10x y=x^2+10x

Video Solution

Step-by-Step Solution

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

 

y=ax2+bx+c y = ax²+bx+c

 

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

c=0 c = 0

 

a is the coefficient of X², here it does not have a coefficient, therefore

a=1 a = 1

 

b=10 b= 10

is the number that comes before the X that is not squared.

 

Answer

a=1,b=10,c=0 a=1,b=10,c=0

Exercise #2

y=2x25x+6 y=2x^2-5x+6

Video Solution

Step-by-Step Solution

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

 

That is,

a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.

Answer

a=2,b=5,c=6 a=2,b=-5,c=6

Exercise #3

What is the value of the coefficient b b in the equation below?

3x2+8x5 3x^2+8x-5

Video Solution

Step-by-Step Solution

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

In the problem, the question was asked: what is the value of the coefficientb b in the equation?

Let's remember the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

ax2+bx+c=0 ax^2+bx+c=0 are :

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That is the coefficientb b is the coefficient of the term in the first power -x x We then examine the equation of the given problem:

3x2+8x5=0 3x^2+8x-5 =0 That is, the number that multiplies

x x is

8 8 Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number8 8 ,

Thus the correct answer is option d.

Answer

8

Exercise #4

What is the value of the coefficient c c in the equation below?

3x2+5x 3x^2+5x

Video Solution

Step-by-Step Solution

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

In the problem, the question was asked: what is the value of the coefficientc c in the equation?

Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

ax2+bx+c=0 ax^2+bx+c=0 are:

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That is the coefficient
c c is the free term - and as such the coefficient of the term is raised to the power of zero -x0 x^0 (Any number other than zero raised to the power of zero equals 1:

x0=1 x^0=1 )

Next we examine the equation of the given problem:

3x2+5x=0 3x^2+5x=0 Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

3x2+5x+0=0 3x^2+5x+0=0 and therefore the value of the coefficientc c is 0.

Hence the correct answer is option c.

Answer

0

Exercise #5

y=x2 y=x^2

Video Solution

Answer

a=1,b=0,c=0 a=1,b=0,c=0

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